Antiskyrmions stabilized at interfaces by anisotropic Dzyaloshinskii-Moriya interaction

TL;DR

This paper demonstrates that antiskyrmions can be stabilized at interfaces in chiral magnets with anisotropic Dzyaloshinskii-Moriya interaction, providing criteria and practical examples for their realization.

Contribution

It establishes the theoretical possibility of antiskyrmions as energy-minimizing states in chiral magnets and identifies specific interface conditions for their stabilization.

Findings

Antiskyrmions can coexist with skyrmions in certain chiral magnets.

Practical criteria for stabilizing antiskyrmions at interfaces are derived.

A Fe/W(110) multilayer is proposed as an experimental platform.

Abstract

Chiral magnets are an emerging class of topological matter harbouring localized and topologically protected vortex-like magnetic textures called skyrmions, which are currently under intense scrutiny as a new entity for information storage and processing. Here, on the level of micromagnetics we rigorously show that chiral magnets cannot only host skyrmions but also antiskyrmions as least-energy configurations over all non-trivial homotopy classes. We derive practical criteria for their occurrence and coexistence with skyrmions that can be fulfilled by (110)-oriented interfaces in dependence on the electronic structure. Relating the electronic structure to an atomistic spin-lattice model by means of density-functional calculations and minimizing the energy on a mesoscopic scale applying spin-relaxation methods, we propose a double layer of Fe grown on a W(110) substrate as a practical example. We conjecture that ultrathin magnetic films grown on semiconductor or heavy metal substrates with $C_{2v}$ symmetry are prototype classes of materials hosting magnetic antiskyrmions.

Equations

$$H=$$

$$-\sum _i\mathrm{B}\cdot {\mathrm{S}}_i+\sum _i{K}_{\perp }({\mathrm{S}}_i\cdot {\hat{\mathrm{e}}}_z{)}^2$$

$$D{\left[\mathbf{m}(\nabla \cdot \mathbf{m})-(\mathbf{m}\cdot \nabla )\mathbf{m}\right]}_z\text{or}D\mathbf{m}\cdot (\nabla \times \mathbf{m}),$$

$$E={\int }_{{\mathbb{R}}^2}\frac{J}{2}|\nabla \mathbf{m}{|}^2$$

$$\mathcal{D}=\frac{1}{A_{\Omega}}\sum_{j\in(\text{n.n.})}\!\!{\boldsymbol{\mathrm{D}}_{0j}\otimes\boldsymbol{\mathrm{R}}_{0j}}=\frac{4\,\text{\DH}}{R}\left(\begin{array}[]{cc}0&R_{y}^{2}\\ -R_{x}^{2}&0\end{array}\right),$$

$$\displaystyle\det\mathcal{D}\,\left\{\begin{array}[]{lll}<0&\text{implies preference of antiskyrmions}\\ >0&\text{implies preference of skyrmions,}\end{array}\right.$$

$$\hslash \frac{d{\mathrm{S}}_i}{dt}=\frac{\partial H}{\partial {\mathrm{S}}_i}\times {\mathrm{S}}_i-\alpha \left(\frac{\partial H}{\partial {\mathrm{S}}_i}\times {\mathrm{S}}_i\right)\times {\mathrm{S}}_i$$

Full text

Chiral magnetic skyrmions 1 are currently subject of intense scientific investigations as they are part of the topological revolution that is currently taking place in condensed matter, with the real-space topology of their magnetic structure playing a leading role. The topological protection of their magnetic structure and particle-like properties with a well-defined topological charge 2 of $Q=-1$ offer good conditions for skyrmions becoming the new information-carrying particles in the field of spintronics 3, and this explains the additional motivation of their current intensive investigation.

Skyrmions in chiral magnets may appear as isolated solitons or condensed in regular lattices. Their stability results from the Dzyaloshinskii-Moriya interaction (DMI) 4; 5, which breaks the chiral symmetry of the magnetic structure. The energy and size is determined by the competition between the Heisenberg, Dzyaloshinskii-Moriya, and Zeeman interaction together with the magnetic anisotropy energy (MAE), here expressed in terms of the spin-lattice model applied to an interface geometry

[TABLE]

with classical spin $\boldsymbol{\mathrm{S}}$ of length one at atomic sites $i,j$ and the corresponding microscopic pair $J_{ij}$, $\boldsymbol{\mathrm{D}}_{ij}$ and on-site $K_{\perp}$ interactions. The DMI results from the spin-orbit interaction and is only non-zero for solids lacking bulk ($\boldsymbol{\mathrm{r}}\nrightarrow-\boldsymbol{\mathrm{r}}$) or structure inversion symmetry ($z\nrightarrow-z$). For applications in spintronics 6; 3; 7 skyrmions stabilized in systems with surface or interface induced DMI seem to be more promising than bulk systems with DMI. For these systems the dipolar interaction between magnetic moments is typically of minor importance and is added as an additional on-site contribution to the magnetic anisotropy energy of the interface, $K_{\perp}$ ($K_{\perp}<0$ means, the easy axis of the magnetization is normal to the interface plane, i.e. along $\hat{\boldsymbol{\mathrm{e}}}_{z}$ direction).

The first example of a topologically non-trivial structure stabilized by the interface DMI was revealed for an Fe monolayer on an Ir(111) substrate 8. Subsequent incremental modifications e.g. adding a monolayer Pd on top of the Fe layer, led to the observation of single skyrmions stablized by interface DMI 9; 10. With respect to future applications, interfaces offer a great variety of options for optimizing and controlling magnetic parameters: Variation of the interface composition 11, of the interface crystal symmetry 12, the film thickness 9, as well as the fabrication of interlayers 13 and multilayers 14.

Taking the micromagnetic view of the DMI, which is valid in the limit of slowly varying magnetic textures, with the prototypical examples of interfacial or cubic DMI,

[TABLE]

respectively, for $\boldsymbol{m}=\boldsymbol{m}(\boldsymbol{\mathrm{r}})$ with $\boldsymbol{\mathrm{m}}(\boldsymbol{\mathrm{R}}_{i})=\boldsymbol{\mathrm{S}}_{i}$, it may not be surprising that the community focuses primarily on the realization of skyrmions ($Q=-1$) rather than antiskyrmions characterized by the topological charge $Q=+1$. The classical Skyrme problem exhibits a reflection symmetry, and the particle and antiparticle with the topological charges $Q=\pm 1$ exist with the same minimal energy 15; 16; 17. This is different for chiral magnetic skyrmions, where this degeneracy is lifted as a consequence of chiral symmetry breaking. In the context of cubic DMI it has been shown that corresponding antiskyrmionic configurations have a strictly higher energy 18. In fact, the densities in (2) can be mapped onto each other by a rigid ($90$ degree) rotation in horizontal spin space and can be considered equivalent. A close inspection of those proofs and simulations which verify that the lowest-energy magnetization configuration of non-trivial topology is attained for $Q=-1$, reveals that a micromagnetic DMI Hamiltonian of the form (2) is assumed with a scalar DM constant $D$, also coined the spiralization.

Different versions of the DMI arise from rigid $O(2)$ transformations applied to the cubic DMI in (2). Some of which, e.g. those transforming to bulk crystals with $S_{4}$ or $D_{2d}$ symmetry, can lead to the stabilization of antiskyrmions rather than skyrmions, as has been recently demonstrated for an acentric tetragonal MnPtPdSn Heusler compound 19 Due to symmetry restrictions, those $O(2)$ transformations are not available for the interfacial DMI (2) and thus antiskyrmions are absent in two-dimensional (2D) or film systems described by interfacial DMI with scalar spiralization. This picture has been confirmed recently by Koshibae and Nagaosa 20, who showed that antiskyrmions are unstable in 2D chiral magnets with scalar $D$, but may be realizable as bubbles in magnets with dipolar interaction.

Thus, more general and interesting are crystal symmetries where the spiralization constant $D$ is replaced by a generic tensor quantity $\mathcal{D}$, which is not included in the $O(2)$ orbit of (2). In this case of anisotropic DMI various scenarios of topological pattern formation including non-symmetric skyrmions and antiskyrmions are possible as shown by means of micromagnetic simulations for combined bulk and interface spin orbit-effects21; 22.

In the framework of this communication we have extended our analysis 18 to two-dimensional chiral systems governed by a micromagnetic energy functional

[TABLE]

with an arbitrary (non-vanishing) spiralization tensor $\mathcal{D}\in\mathbb{R}^{2\times 2}$ emerging from (1). Here, $\mathcal{D}:\mathcal{L}(\boldsymbol{m})$ denotes the contraction with the chirality tensor $\mathcal{L}(\boldsymbol{m})=\nabla\boldsymbol{m}\times\boldsymbol{m}$ over $x,y$ indices in real and spin space (see Supplementary Note 1 for details). The analysis provides a precise criterion for the optimality of antiskyrmions versus skyrmions and reveals the diversity of possible chiral phenomena including coexistence of both entities in chiral magnets.

The topologically nontrivial magnetization fields are vectorial quantities, assuming values in a multidimensional manifold. Thus, it is conceivable that they are characterized by more than one topological index.

For skyrmionic configurations with a well-defined skyrmion center $\boldsymbol{\mathrm{r}}_{0}$ where $m_{z}^{2}=1$, it is instructive to take into account the index $v$ of the horizontal magnetization field $m_{\scriptscriptstyle\parallel}=(m_{x},m_{y})=\sqrt{1-m_{z}^{2}}e^{\rm{i}\varphi}$ at $\boldsymbol{\mathrm{r}}_{0}$, given by $v=\frac{1}{2\pi}\oint_{\Gamma}\nabla\varphi\cdot\mathrm{d}\boldsymbol{\mathrm{r}}$, which assumes the same integer value for every oriented Jordan curve $\Gamma$ enclosing $\boldsymbol{\mathrm{r}}_{0}$. In this case, $v$ can be considered as a secondary topological charge and the defining index to distinguish between skyrmion ($v=1$) and antiskyrmion ($v=-1$) independently of the background state. The index $v$ is also referred to as the $\mathbb{S}^{1}$ winding number in comparison to the topological charge 2, $Q$, which is the $\mathbb{S}^{2}$ winding number. Being independent of $m_{z}$, the $\mathbb{S}^{1}$ winding number is independent of the direction of the field polarization $\pm\hat{\boldsymbol{\mathrm{e}}}_{z}$.

The DM vector $\boldsymbol{\mathrm{D}}_{ij}$ imposes a handedness on the magnetic structure relating the rotation of the spin $\boldsymbol{\mathrm{S}}$ about the chirality vector $\boldsymbol{\mathrm{c}}_{\chi}(\hat{\boldsymbol{\mathrm{R}}}_{ij})=\boldsymbol{\mathrm{S}}_{i}\times\boldsymbol{\mathrm{S}}_{j}$ to the displacement of the spin along the direction $\hat{\boldsymbol{\mathrm{R}}}_{ij}=(\boldsymbol{\mathrm{R}}_{j}-\boldsymbol{\mathrm{R}}_{i})/|\boldsymbol{\mathrm{R}}_{j}-\boldsymbol{\mathrm{R}}_{i}|$ of the bond between the atoms $i$ and $j$. According to our sign convention in (1), the energy is minimized if $\boldsymbol{\mathrm{c}}_{\chi}(\hat{\boldsymbol{\mathrm{R}}}_{ij})\parallel\boldsymbol{\mathrm{D}}_{ij}$. We speak of right- (left-) handed Bloch-type skyrmions if $\boldsymbol{\mathrm{c}}_{\chi}(\hat{\boldsymbol{\mathrm{R}}}_{ij})\cdot\hat{\boldsymbol{\mathrm{R}}}_{ij}>0\,(<0)$, and of Néel-type skyrmions if $\boldsymbol{\mathrm{c}}_{\chi}(\hat{\boldsymbol{\mathrm{R}}}_{ij})\cdot\hat{\boldsymbol{\mathrm{R}}}^{\perp}_{ij}>0\,(<0)$, where $\hat{\boldsymbol{\mathrm{R}}}^{\perp}$ is an arbitrary vector of the positive quadrant of the orthogonal complement of $\hat{\boldsymbol{\mathrm{R}}}$, e.g., formed by the vectors $\hat{\boldsymbol{\mathrm{e}}}_{z}\times\hat{\boldsymbol{\mathrm{R}}}$ and $\hat{\boldsymbol{\mathrm{e}}}_{z}$.

Although the handedness is not a topological property and thus does not affect topological indices nor the stability of skyrmions, it connects to interesting physical properties. As discussed in more detail below skymions and antiskyrmions are also distinct in their handedness properties.

A double layer Fe on W(110) exhibits a $C_{2v}$ symmetry and is an example with non-zero tensorial components of $\mathcal{D}$. We show in this communication by means of vector-spin density functional theory (DFT) calculations that its electronic structure leads to long-range and microscopically anisotropic DM vectors that add up to tensorial elements of $\mathcal{D}$ such that the antiskyrmion becomes the non-trivial low-energy magnetization. We confirm the stability of antiskyrmions by energy minimization of Eq. (1) on a mesoscopic scale employing atomistic spin-dynamics.

I Results

Symmetry Analysis. What may be surprising is that interface stabilized skyrmions have exclusively been explored for (111) oriented interfaces exhibiting a $C_{3v}$ symmetry with the exception of Mn/W(100), which has a $C_{4v}$ symmetry 13. Such high-symmetry interfaces may only support Néel- or hedgehog-type monochiral skyrmions of positive $\mathbb{S}^{1}$ winding number, $v=1$, with the magnetization field $\boldsymbol{\mathrm{m}}(\boldsymbol{\mathrm{r}})=m_{\rho}(\rho)\hat{\boldsymbol{\mathrm{e}}}_{\rho}+m_{z}(\rho)\hat{\boldsymbol{\mathrm{e}}}_{z}$ when expressed in a cylindrical coordinate system $(\rho,\varphi,z)$ and micromagnetic theory is applied. It is characterized by a fixed sense of cycloidal rotation of magnetic moments from the core outwards independent of the radial direction $\hat{\boldsymbol{\mathrm{e}}}_{\rho}$. The sense of rotation can be either clock- or counter-clockwise consistent with the chiral symmetry breaking. It becomes a monopole hedgehog when mapped onto a sphere, see Fig. 1(a),(c), and (e).

Such monochiral skyrmions are formed when the direction of the DM vector relative to the bond direction does not change significantly for a pair of coupled spins in different crystallographic directions. Examples are systems with $C_{3v}$ or $C_{4v}$ symmetry, which appear typically at (111) or (001) oriented interfaces and surfaces of bcc and fcc crystals, respectively. This goes all the way back to symmetry arguments for DM vectors, also known as Moriya rules 5. When applied to these high-symmetry surfaces, they determine uniquely the in-plane components of the direction of the DM vectors, $\hat{\boldsymbol{\mathrm{e}}}_{\parallel\text{DM}}$, to point perpendicular to the bond connecting the two interacting spins, see in-plane components of the nearest-neighbour (n.n.) DM vectors (blue arrow) $\hat{\boldsymbol{\mathrm{e}}}_{\parallel\text{DM}}\parallel-\hat{\boldsymbol{\mathrm{e}}}_{\varphi}$ in Fig. 2(a) and (b). Only one scalar parameter $D$ controlling the sign and absolute value along $\hat{\boldsymbol{\mathrm{e}}}_{\parallel\text{DM}}$ may vary in dependence on the electronic structure. Furthermore, the mirror symmetries of these surfaces force the DM vectors to have the same $\varphi$-component $D_{\varphi}$ for all symmetry-equivalent atom pairs. Thus, variations of the magnetization along a radial path of any direction $\hat{\boldsymbol{\mathrm{e}}}_{\rho}$ will have the same preferred axis of rotation, $\boldsymbol{\mathrm{c}}_{\chi}\parallel-\hat{\boldsymbol{\mathrm{e}}}_{\varphi}$ (highlighted by the smaller green arrows) when going along different crystallographic directions and therefore only monochiral skyrmions can be formed by the DM interaction.

The antiskyrmion (see Fig. 1(b),(d), and (f)), on the other hand, is the simplest form of a multichiral skyrmion 23, which is described by a polar angle $\varphi$-dependent magnetization field $\boldsymbol{\mathrm{m}}(\boldsymbol{\mathrm{r}})=m_{\rho}(\rho,\varphi)\hat{\boldsymbol{\mathrm{e}}}_{\rho}+m_{\varphi}(\rho,\varphi)\hat{\boldsymbol{\mathrm{e}}}_{\varphi}+m_{z}(\rho)\hat{\boldsymbol{\mathrm{e}}}_{z}$, with negative $\mathbb{S}^{1}$ winding number, $v=-1$, characterized by different rotational senses along different radial directions away from the core (cf. Fig. 1(f)), thus showing multiple chiralities. It can be understood naively as an addition of a quadrupolar field to the monopole of the skyrmion (see Supplementary Note 2 and Supplementary Movie 2). Thus, the physical magnetization space has an orientational energy dependence relative to the lattice, a property which may add significance to the role of antiskyrmion applications. Due to the multichirality of the magnetic texture, the winding of $m_{\rho}$ along certain directions $\hat{\boldsymbol{\mathrm{e}}}_{\varphi}$ costs energy by the DMI and makes the antiskyrmion seem unfavourable over the skyrmion.

To form such multichiral structures stabilized by DMI, the DMI evidently needs to show a strong directional dependence. In films an anisotropic DMI is allowed by symmetries lower than $C_{3v}$. This is schematically sketched in Fig. 2(c) and (d) for the example of a surface with a centered-rectangular lattice exhibiting $C_{2v}$ symmetry. In this case, the number of symmetry operations is so low that the in-plane direction of the DM vectors, $\hat{\boldsymbol{\mathrm{e}}}_{\parallel\text{DM}}$, is not anymore uniquely determined by symmetry. Here, the remaining two mirror symmetries constrain only one component of the DM vector, namely the $D_{z}$-component to vanish, but the DM vectors are free to point into any in-plane direction.

We recall that distance vectors, $\boldsymbol{\mathrm{R}}_{ij}$, between pairs of atoms $(i,j)$ form shells of symmetry equivalent vectors of equal length generated by the symmetry operations $\{\mathcal{R}\}$ mapping atom $j$ onto atom $j^{\prime}$ and consequently the pseudovector $\boldsymbol{\mathrm{D}}_{ij^{\prime}}=\det(\mathcal{R})\mathcal{R}\cdot\boldsymbol{\mathrm{D}}_{ij}$ (see e.g., blue arrows in Fig. 2(d), which all point along the bond-direction).

Magnetostatic energy difference between skyrmion and antiskyrmion. As a consequence of the different magnetisation densities, skyrmions and antiskyrmions embedded in a ferromagnetic background experience a different magnetostatic self-energy $E_{\textrm{mag}}$. For an infinite 2D film, $E_{\textrm{mag}}$ (averaged by the film thickness) decomposes to leading orders into (i) a shape anisotropy energy, $E_{\textrm{mag}}^{\perp}$, and (ii) a magnetic film charge term $E_{\textrm{mag}}^{\scriptscriptstyle\parallel}$ 25. The first term depends solely on $m_{z}^{2}$, is independent of the index $v$ and accounts for the ferromagnetic dipolar interaction energy included already in the ferromagnetic anisotropy constant $K_{\perp}$. The second term is proportional to the film thickness $t$, depends on the magnetic film charge $\sigma=(\nabla\cdot m_{\scriptscriptstyle\parallel})$ and is sensitive to the index $v$ and the type of skyrmions characterized by the chirality vector $\boldsymbol{\mathrm{c}}_{\chi}(\hat{\boldsymbol{\mathrm{R}}}_{ij})$. $E_{\textrm{mag}}^{\scriptscriptstyle\parallel}$ as a function of $v$ and $\boldsymbol{\mathrm{c}}_{\chi}$ is maximal for Néel-type skyrmions, zero for Bloch-type skyrmions and precisely in the middle for axisymmetric antiskyrmions, see Supplementary Note 3.

Micromagnetic Arguments for skyrmion formation. The additional degree-of-freedom for the direction $\hat{\boldsymbol{\mathrm{e}}}_{\parallel\textnormal{DM}}$ may allow for the formation of antiskyrmions over monochiral skyrmions. We will verify this by inspecting in the following the winding of the magnetic structure along $\hat{\boldsymbol{\mathrm{e}}}_{\rho}$ from inside out in dependence on $\varphi$ assuming only n.n. interaction for two extreme cases of DM vector directions for a system with $C_{2v}$ symmetry: (i) the n.n. DM vectors point perpendicular to the bond as shown in Fig. 2(c) (see blue arrows). In this case the spiralization tensor for 2D systems 24 takes the form

[TABLE]

where $\boldsymbol{\mathrm{R}}_{0j}=(R_{x},R_{y})$ is the distance vector to the n.n. atom, $R=\lvert\boldsymbol{\mathrm{R}}_{0j}\rvert$, $\text{\DH}=\lvert\boldsymbol{\mathrm{D}}_{0j}\rvert$ and $A_{\Omega}$ is the area per surface atom.

The effective DM vector in the micromagnetic sense, along the direction of the radial coordinate $\hat{\boldsymbol{\mathrm{e}}}_{\rho}=(\cos(\varphi),\sin(\varphi))^{\mathrm{T}}$ is given by $\boldsymbol{\mathrm{D}}_{\chi}=A_{\Omega}\mathcal{D}\,\hat{\boldsymbol{\mathrm{e}}}_{\rho}$ and indicated in Fig. 2 by thin arrows for four directions of $\hat{\boldsymbol{\mathrm{e}}}_{\rho}$ along the four crystallographic $\pm x$- and $\pm y$-axes, respectively. Their green colour indicates that they result all in the same Néel-type radial chirality, $\mathcal{C}_{N}$, which we define as projection of $\boldsymbol{\mathrm{D}}_{\chi}$ onto $\hat{\boldsymbol{\mathrm{e}}}_{\varphi}$, $\mathcal{C}_{N}(\varphi)=(\mathcal{D}\hat{\boldsymbol{\mathrm{e}}}_{\rho})_{\varphi}$. For the case sketched in Fig. 2(c) it is always negative, $\mathcal{C}_{N}=-\frac{4\,\text{\DH}}{R}\,\left(R_{x}^{2}\cos^{2}\varphi+R_{y}^{2}\sin^{2}\varphi\right)$, irrespective of the radial direction $\hat{\boldsymbol{\mathrm{e}}}_{\rho}$, i.e. the green arrows of Fig. 2(c) show in the $-\hat{\boldsymbol{\mathrm{e}}}_{\varphi}$ direction. The independence of $\mathcal{C}_{N}$ on the radial direction is analogous to interfaces with $C_{3v}$ and $C_{4v}$ symmetry and stabilizes monochiral skyrmions only (see Fig. 2(e)). Since the micromagnetic energy function is minimized if the radial vector chirality of the spin-structure $\boldsymbol{\mathrm{c}}_{\chi}(\hat{\boldsymbol{\mathrm{e}}}_{\rho})$ points parallel to $\boldsymbol{\mathrm{D}}_{\chi}$, the negative radial chirality $\mathcal{C}_{N}$ corresponds to a left- or counter-clockwise magnetic structure along direction $\hat{\boldsymbol{\mathrm{e}}}_{\rho}$ as shown in Fig. 2(e).

(ii) On the contrary, if the n.n. DM vectors point parallel to the bonds as in Fig. 2(d), the off-diagonal entries of (4) change to $-4\text{\DH}R_{x}R_{y}/R$ and the Néel-type chirality $\mathcal{C}_{N}(\varphi)=\frac{4\,\text{\DH}\,R_{x}R_{y}}{R}\left(\sin^{2}\varphi-\cos^{2}\varphi\right)$ takes positive and negative values, depending on the radial or crystallographic direction of $\hat{\boldsymbol{\mathrm{e}}}_{\rho}$, as indicated by red and green colour, respectively. It is now evident, that multichiral antiskyrmions are stabilized by such a DM-field.

We stress that in general all directions of the DM vector are possible. They are finally system dependent and determined by the details of the electronic structure. Even when the DM vectors deviate considerably from the direction of the bond, multichiral spin-textures remain lower in energy than monochiral ones as we show in the Supplementary Note 4.

A whole class of systems which possess such a $C_{2v}$ symmetry are magnetic films on the (110) surfaces of bcc or fcc crystals. Selecting a heavy metal substrate may result in a strong DM interaction 26. In this context the prototype experimental system is certainly an Fe double layer on W(110) 27; 28; 29; 30. It was shown that the ground state of the system is a ferromagnet with an out-of-plane easy axis, which is favourable for skyrmion stabilization 31 and it shows a DMI, not strong enough to establish a chiral ground state, but strong enough to stabilize Néel-type domain walls of unique rotational sense 32; 29; 30. Interestingly, a sign change in the micromagnetic DMI along different high-symmetry directions was predicted theoretically for one and two layers of Fe on W(110) 32.

Mathematical analysis of (anti-)skyrmion stability and coexistence. Relating a low-symmetry interface to a two-dimensional spiralization tensor we could provide arguments that make the stabilization of antiskyrmions plausible. This challenges the arguments of ref. 18, which identify skyrmions as local energy minimizer within the nontrivial topological sector $Q=-1$, i.e. with $\mathbb{S}^{1}$ winding number $v=1$. In the Supplementary Note 1 we generalized the arguments to arbitrary two-dimensional spiralization tensors $\mathcal{D}$. Central results are the following:

[TABLE]

see Theorem 1 and 2 in Supplementary Note 1. In the anisotropic case, where the $\mathcal{D}$ tensor possesses different singular values $0<D_{1}<D_{2}$, skyrmions and antiskyrmions coexist, see Theorem 3 in Supplementary Note 1, with approximately equal energies if $D_{1}\approx 0$, i.e. $\det\mathcal{D}\approx 0$, marking a transition from a skyrmionic to an antiskyrmionic phase.

First-principles calculations. So far, most DM parameters calculated from ab initio were either limited to n.n. interactions 33; 34 or to the spiralization 10; 32; 31; 24; 35, the DM strength in the micromagnetic approach, an approximation where most information about the underlying lattice structure is averaged out. Here, we go beyond these limitations and move to a more realistic description, namely the atomistic spin-lattice model (1). It includes also the pair-wise DM interactions beyond the n.n. approximation, which we directly calculate by means of density functional theory (DFT) (see Methods) both for atoms within the surface and interface layer but also in-between both representing the intra- and interlayer spin-spin interactions, respectively. Those pair-wise DM vectors together with the exchange constants and the on-site anisotropy are then used in spin-dynamics simulations to investigate the stability of skyrmions and antiskyrmions in the system.

Our DFT results for the exchange constants $J_{ij}$ witness a strong ferromagnetic behaviour both within each layer separately (n.n. values $J^{S}_{01}=9.07$ meV and $J^{I}_{01}=14.24$ meV for the surface and interface layer, respectively) and especially between both layers ($J^{SI}_{01}=42.19$ meV). Going beyond the n.n. approximation allows for a far more accurate description of the physical system (the complete set of values is diagrammed in Supplementary Fig. 4, see Supplementary Note 5).

The calculated intralayer DM vectors are depicted in Fig. 3(a) and (b). In the surface layer the first n.n. DM interaction dominates over all other components, while in the interface layer also the DM vectors of the second n.n. are of significant size, but of opposite direction. This oscillatory behaviour can also be seen in Fig. 3(d), where the distance dependence of $\operatorname{sign}{(\boldsymbol{\mathrm{D}}_{ij})_{\varphi}}\lvert(\boldsymbol{\mathrm{D}}_{ij})_{\scriptscriptstyle\|}\rvert$, i.e. with respect to $|\boldsymbol{\mathrm{R}}_{i}-\boldsymbol{\mathrm{R}}_{j}|$ is diagrammed. $(\boldsymbol{\mathrm{D}}_{ij})_{\scriptscriptstyle\|}$ is the component of the DM vector parallel to the surface and $\operatorname{sign}{(\boldsymbol{\mathrm{D}}_{ij})_{\varphi}}$ contributing to the preferred Néel-type handedness of the spin texture ($\operatorname{sign}{(\boldsymbol{\mathrm{D}}_{ij})_{\varphi}}\rightarrow\operatorname{sign}{\mathcal{C}_{N}}$. In Fig. 3(c) the sum of both intralayer DM interactions are superimposed on the surface plane. One can see, that the contributions almost compensate each other (see e.g., the first nearest neighbours), and that this compensation leads to a reduction of the absolute value of the effective n.n. interaction to the point that the next shells become significant. This shows again the importance to go beyond a n.n. approximation to describe this system.

Fig. 3(e) displays the highly anisotropic nature of the DM vectors of this system expressed in terms of wavevectors in reciprocal space, $\boldsymbol{\mathrm{D}}(\boldsymbol{\mathrm{q}})$, and plotted for an area around the $\bar{\Gamma}$-point (ferromagnetic state) of the 2D Brillouin zone (BZ), an appropriate description for films with crystal periodicity. The slope of $(\boldsymbol{\mathrm{D}}(\boldsymbol{\mathrm{q}}))_{\varphi}$ close to the $\bar{\Gamma}$-point along the different crystallographic directions provides the off-diagonal elements of the spiralization tensor. For 2Fe/W(110) we obtain $\mathcal{D}_{12}=-7.85$ meV/nm ($-1.26$ pJ/m) and $\mathcal{D}_{21}=-6.20$ meV/nm ($-0.99$ pJ/m) 36, respectively, which is in good agreement with the calculated values in ref. 32. The diagonal elements $\mathcal{D}_{11}=\mathcal{D}_{22}=0$ are zero due to $C_{\text{2v}}$ symmetry and the singular values of $\mathcal{D}$ are $D_{1}=|\mathcal{D}_{21}|$ and $D_{2}=|\mathcal{D}_{12}|$, respectively. The determinant $\det\mathcal{D}=-\mathcal{D}_{12}\mathcal{D}_{21}<0$, i.e. the micromagnetic antiskyrmion condition (7), which predicts antiskyrmions as the non-trivial topological magnetization soliton of lowest energy, is satisfied. In Fig. 3(e), the different preferred rotation axes, $\boldsymbol{\mathrm{c}}_{\chi}(\boldsymbol{\mathrm{q}})\propto\boldsymbol{\mathrm{D}}(\boldsymbol{\mathrm{q}})$, along different crystallographic directions can be clearly seen. However, it also becomes obvious that the preferred rotation axis changes for different periods of spin-spirals even if they propagate along the same direction. With this observation micromagnetic theory might come at its limit. Thus, despite the fact that the micromagnetic criterion for antiskyrmion stability is satisfied, it is advisable to analyse beyond the micromagnetic theory, whether these $\boldsymbol{\mathrm{D}}$ vectors can lead to the formation of stable antiskyrmions.

We turn briefly to the MAE, $K_{\perp}$ in (1), exhibiting a total out-of-plane easy axis with $K_{\perp}=-0.11$ meV/Fe-atom. In addition to the magnetocrystalline anisotropy ($-0.25$ meV), the dipole-dipole interaction is included here as an effective on-site anisotropy ($0.14$ meV).

Atomistic spin-dynamics simulations. To analyse the stability of skyrmions and antiskyrmions in 2Fe/W(110) on a mesoscopic scale we performed atomistic spin-dynamics simulations at zero temperature using the extended first-principles Heisenberg model (1) with parameters determined above (see Methods).

While our simulations confirm the ferromagnetic state to be the ground state, antiskyrmions (see Fig. 4 for a top view of the resulting structure) appear as stable states for the interaction parameters describing 2Fe/W(110). They can be found starting from different initial configurations, e.g., from antiskyrmions with a broad range of diameters, irrespective of their spatial orientation, as well as from random configurations. We also investigated the stability of the antiskyrmion with respect to the different number of shells of neighbours resolved in the first-principles calculations and subsequently taken into account in the spin-dynamics simulations. Stability of antiskyrmions was found for all tested numbers of shells, however, the exact diameter of the antiskyrmion might vary. Furthermore, for some numbers of shells, a small magnetic field was required to stabilize the antiskyrmion, while for most cases, the antiskyrmion was stable even at zero magnetic field. However, in all tested cases the required field was lower than $150$ mT, equivalent to about $0.02$ meV/Fe-atom, which lies clearly within the limit of first-principles calculations.

The anisotropy of the micromagnetic $\mathcal{D}$-tensor, $2|{D}_{1}-{D}_{2}|/({D}_{1}+{D}_{2})$ amounts to 23%. Although the anisotropy is small, our micromagnetic theory suggests a possible coexistence of skyrmions with antiskyrmions. Starting the atomistic spin-dynamics simulations from any chiral skyrmion, we observe their collapse into the ferromagnetic state if no external magnetic field is applied. Applying an magnetic field parallel to the core of the skyrmion, we observe in our simulations that at about $700$ mT (the exact value depends again on the number of shells which are taken into account) the skyrmion no longer collapses. Instead, the skyrmion expands towards the boundary of our simulated lattice resulting in a ferromagnetic state aligned parallel to the magnetic field. Fine-tracking the stability of skyrmions with respect to the external magnetic field we can say that within a field resolution of $1$ mT no stable skyrmion was found. A further analysis of the role of different DM energy contributions (intra- and interlayer coupling) on the stabilization of antiskyrmions can be found in the Supplementary Note 6.

II Discussion

The interplay of energy and topology resulting in the optimality of certain topological states in micromagnetics is revealed by general principles of mathematical analysis. In the context of 2D systems where DMI is represented by a $2\times 2$ spiralization tensor $\mathcal{D}$, it follows that the energetically selected homotopy class (skyrmion versus antiskyrmion) is determined by the orientation of $\mathcal{D}$, i.e. the sign of $\det\mathcal{D}$ (see Theorem 1 in Supplementary Note 1). Chiral features of the skyrmion texture are determined by the tensorial components of $\mathcal{D}$. E.g. right- (left-) handed Néel-type skyrmions typical for interfaces are obtained if $\operatorname{sign}\left(\mathcal{D}_{21}-\mathcal{D}_{12}\right)=1\ (-1)$ provided $\mathcal{D}_{21}\neq\mathcal{D}_{12}$.

The prototypical forms of DMI arising in ultra-thin films or from 2D reductions of the cubic B20 type satisfy $\det\mathcal{D}=D^{2}>0$ where $D\not=0$ is the DMI constant. As is known 18, chiral skyrmions with $Q=-1$ occur as local energy minimizers within the field polarized regime $BJ>D^{2}$. The crucial observation is that the least energies $E_{Q}$ in the topological sectors of charge $Q=\pm 1$ are separated: $E_{-1}<4\pi J$ while $E_{1}=4\pi J$, where the energy quantum $4\pi J$ is the threshold for the collapse of a topological entity. Consequently, antiskyrmionic configurations may release energy at the expense of forming a point singularity.

Tuning the system towards $\mathcal{D}$ with negative determinant, as in the example of an Fe double layer on W(110), lowers the least energy in the topological sector $Q=1$ below $Q=-1$, i.e. preference of antiskyrmions. A particularly interesting situation arises in the limit case of vanishing determinant, i.e. the presence of effectively only one DM vector, where our analysis predicts the coexistence of skyrmions and antiskyrmions with equal energies. Coexistence, however, is not exclusive to this extreme situation. Whenever the singular values $D_{1}$ and $D_{2}$ of $\mathcal{D}$ differ, the least energies in both topological sectors $Q=\pm 1$ are subcritical $E_{Q}<4\pi J$. This allows for the occurrence of both entities, while $\det\mathcal{D}$ serves as a measure of the difference of their energies.

The magnetostatic energy is not able to change the handedness of the skyrmion, but can be used to tune the energy difference between the skyrmion and antiskyrmion on a fine energy scale as function of the film thickness.

Since the $\mathcal{D}$ tensor is a measurable as well as computable quantity, the sign and value of the determinant serves as a simple but powerful criterion for the occurrence and stability of skyrmions and antiskyrmions as well as an effective property descriptor to navigate the search for new materials. It serves also as classification scheme of chiral magnets into isotropic rank-three DMI bulk and rank-two DMI film magnets, with a DMI described by a single constant, the spiralization, for which antiskyrmions are stable only for bulk crystals with certain point group symmetries. Then, we have the anisotropic rank-two DMI film magnets, where skyrmions and antiskyrmions can coexist, while the sign of $\det\mathcal{D}$ determines which of the two has the lower energy. Finally, zero determinant indicates a rank-one DMI material, for which skyrmions and antiskyrmions have the same energy.

Our theoretical predictions are consistent with the calculations of Güngordü et al. 22 observing the occurrence of skyrmion or antiskyrmion lattices depending on details of effective spin-orbit models. Our analysis now provides a precise selection criterion in the case of general $2\times 2$ spiralization tensors. Coexistence of isolated skyrmions and antiskyrmions for anisotropic DMI is an entirely new theoretical aspect in the field of chiral skyrmions.

Combining DFT with atomistic spin-dynamics simulations, in this paper we showed for a double layer Fe on W(110) that the interplay of Heisenberg and anisotropic Dzyaloshinskii-Moriya exchange with uniaxial magnetocrystalline anisotropy and an at most modest external magnetic field results in fact in the stabilization of antiskyrmions. The spatially anisotropic DMI-vectors, $\boldsymbol{\mathrm{D}}_{ij}$, enabled by the $C_{2v}$ symmetry sum up to a spiralization tensor $\mathcal{D}$, rather than a spiralization scalar $D$, with negative determinant consistent with the micromagnetic antiskyrmion criterion. The tensor is surprisingly isotropic as evidenced by the singular values $D_{1}\approx D_{2}$ and no coexistence of skyrmion and antiskyrmion was found by means of spin dynamics. Details of the electronic structure lead to pair interactions of the DMI of competing sign, showing that a nearest neighbour approximation is insufficient to resolve stability, shape, orientation and fine-structure of the antiskyrmion. As seen from Fig. 4 the symmetry axis of the antiskyrmions matches with the underlying crystal lattice in that the counter-clockwise rotation of the magnetic moments follows the long axis of the unit cell and the clockwise the short one.

Considering that heavy transition-metal substrates 31 like W contribute most to the DMI and that their complex Fermi surfaces 37 lead to rather anisotropic $\boldsymbol{\mathrm{D}}_{ij}$ parameters, together with the $C_{2v}$ symmetry we consider 2Fe/W(110) as a prototypical example of a much wider class of magnetic films and heterostructures that host antiskyrmions rather than skyrmions as topologically nontrivial stable states. Other members of this class are certainly those with (110) oriented interfaces between $3d$ and $5d$ transition-metals. In fact, an anisotropic DMI was recently measured in a thin epitaxial Au/Co film on W(110) 38, but this system favored elliptical skyrmions instead of anti-skyrmions. Other candidates are $3d$ metals on semiconductor (100) and (110) surfaces, e.g., Fe on Ge or GaAs, exhibiting $C_{2v}$ and $C_{s}$ symmetry, respectively.

In this communication we focused on interface stabilized antiskyrmions typical for thin magnetic films and heterostructures, but our arguments apply also to bulk systems with broken inversion symmetry and space groups, for which the effective spiralization tensor is described by a $2\times 2$ matrix rather than a scalar. This motivates the synthesis of new magnetic materials e.g., with $C_{2v}$ symmetry on the basis of quaternary selenides such as InVSe2O839. When typical chiral bulk magnets, e.g., B20 alloys, with their lack of bulk inversion symmetry ($\boldsymbol{\mathrm{r}}\nrightarrow-\boldsymbol{\mathrm{r}}$) with a DMI described by a spiralization scalar $D$ meet the inversion asymmetry ($z\nrightarrow-z$) of surfaces, films and heterostructures, we expect always a non-zero $\mathcal{D}$-tensor. Thus, antiskyrmions can emerge at interfaces of skyrmion carrying bulk phases. To complete the discussion, antiskyrmions in films with symmetries resulting in a scalar spiralization do not exist, but can be found in bulk crystals with space groups $D_{2d}$ and $S_{4}$. The most promising systems to host antiskyrmions seem to be magnetic Heusler alloys (e.g., Mn2RhSn 40), chalcopyrites (e.g., CuFeS2 41), stannites (e.g., Cu2FeSnSe4 42) or kesterites (e.g., Cu2Mn1-xCoxSnS443).

Since the topological charge is related to the gyrovector and the Magnus force acting on the antiskyrmion, we expect an antiskyrmion Hall effect. Thus, our work prompts the exploration of the dynamical, dissipative and transport properties related to the additional orientational stabilization of the antiskyrmions with respect to the underlying lattice in comparison to the monochiral skyrmions.

This work motivates also the design of materials that can host skyrmions and antiskyrmions simultaneously (case $0<D_{1}<D_{2}$). This opens an exciting perspective to investigate the interaction of particle and antiparticle of different energies in the spirit of the skyrmion to antiskyrmion interaction in dipolar magnets 20 or frustrated magnets 44 and the conditions of mixed ordered lattices and phases. It should be explored in how far the tunnelling mixing magneto-resistance 45; 46 (TXMR) effect can be used to discern electrically skyrmions from antiskyrmions. If we optimize materials parameters such that skyrmions as well as antiskyrmions become stable, conditions of mixed ordered lattices and phases may be met. We expect that the topological charge density of an antiskyrmion produces an emergent magnetic field opposite to that of skyrmions for the same material. These different emergent fields give rise to topological orbital moments of opposite sign that can be exploited to discriminate the different skyrmion-antiskyrmion phases spectroscopically using soft X-ray magnetic circular dichroism 47 (XMCD). These mixed phases are lattices of staggered magnetic fields. The topological Hall effect (THE) of these lattices would be an exciting topic to study.

Rank-one materials should be particularly interesting for information storage and processing. One may envisage a creation process of a skyrmion-antiskyrmion pair out of the trivial FM state or a domain wall keeping the total $\mathbb{S}^{1}$ and $\mathbb{S}^{2}$ winding numbers, $v$ and $Q$, zero. They allow for an extension of the skyrmion race track idea 3, where the information is encoded in the relative positions or time sequences, respectively, of the skyrmions along the track, to the skrymion-antiskyrmion race track memory 48, where the binary information is encoded in the sequence of skyrmions and antiskyrmions, which are expected to be read out distinctly. Upon further investigations of skyrmion-antiskyrmion interactions in constrictions, rank-one materials may be the ideal host to extend the concepts of the skyrmion logic gates 49 to magnetic logic gates in which skyrmions and antiskyrmions are the elementary particles for binary operations.

III Methods

Mathematical analysis. Rather than attempting to solve the equilibrium equations directly, we examine the micromagnetic energy landscape over different homotopy classes. Chiral (anti-)skyrmions arise as relative energy minimizers, which are stable with respect to arbitrary perturbations in the configuration space. The proof of (co-)existence is based on variational arguments combining constructive upper and ansatz-free lower energy bounds of Bogomolny type. The bounds particularly capture the energetic optimality of antiskyrmionic versus skyrmionic configurations (or vice versa) leading to the micromagnetic criterion (7). Crucially in the context of anisotropic DMI, the approach does not rely on any symmetry assumption. Our preliminary symmetry analysis exploits independent $O(2)$ transformations in spin and real space and a singular value decomposition of the spiraliztion tensor (rather than an additive decomposition22) to identify a canonical form of DMI. Details are provided in Supplementary Note 1.

First-principles calculations. We performed vector-spin density functional theory (DFT) calculations using the film version of the full-potential linearized augmented plane wave method (FLAPW) as implemented in the FLEUR code 50. By using the FLEUR code, we get access to the total energy of non-collinear magnetic structures and spin-spirals both with and without spin-orbit coupling, which allows to obtain the parameters entering Eq. (1). For details on the computations and choice of exchange correlation potential we refer to Supplementary Note 5.

To calculate the pair-wise DM vectors from DFT, we extended the derivation of interatomic exchange interactions in ferromagnets by Ležaić et al. 51 (which we also used here to calculate the exchange constants $J_{ij}$) to the interatomic DMI parameters $\boldsymbol{\mathrm{D}}_{ij}$. These have then been evaluated by means of the FLEUR code carrying out coned spin-spirals calculations for a grid of wavevectors $\boldsymbol{\mathrm{q}}$ with a small cone angle $\theta$ employing the force theorem (for details see Supplementary Note 5). The results have been cross-checked against independent Korringa-Kohn-Rostocker Green-function (KKR-GF) calculations and we obtained qualitative and quantitative agreement (see Supplementary Note 7).

Atomistic spin-dynamics simulations. In order to analyse the stability of skyrmions and antiskyrmions we have relaxed approximate and random spin configurations on a lattice with ($150\times 150$) spins in each layer of the Fe double layer at zero temperature according to the Landau–Lifshitz-Gilbert equation of spin dynamics:

[TABLE]

using the extended Heisenberg model (1), where $H$ is the Hamiltonian, $\boldsymbol{\mathrm{S}}_{i}$ is the spin of unit-length at site $i$ and $\alpha$ is the damping parameter. To achieve fast relaxation from initial distortions of the spin system along the physical path towards lower energy, we have used a range of $\alpha=0.7$ to $0.1$ and time steps ranging from $0.3$ to $3$ fs, respectively. We included the interaction parameters of the neighbours shown in Fig. 3 and in the Supplementary Fig. 4 (see Supplementary Note 5) for the pair-wise contributions including up to 21 shells with in total 71 pairs per atom for both the intralayer and the interlayer coupling. We have carried out simulations for periodic boundary conditions, so as to avoid influence of the boundary on the stabilization of the antiskyrmion. The equation of motion was integrated by applying an efficient and robust semi-implicit numerical method with built-in angular momentum conservation 52, as implemented in Spirit 53. States, which are stabilized this way in spite of the different initial perturbations or distortions, we experienced as local minima in the energy landscape described by the Hamiltonian.

IV Data availability

The data that support the findings of this study are available from the corresponding authors upon request.

V Acknowledgments

We would like to thank Miriam Hinzen, Frank Freimuth, Yuriy Mokrousov, and Hannes Jónsson for fruitful discussions. We gratefully acknowledge computing time on the JURECA supercomputer provided by the Jülich Supercomputing Centre (JSC). B.Z. and S.B. acknowledge funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement number 665095 (FET-Open project MAGicSky). G.P.M. acknowledges funding from the Icelandic Research Fund (grant no. 152483-052). Ch.M. acknowledges funding from Deutsche Forschungsgemeinschaft (DFG grant no. ME 2273/3-1). Ch.M. and S.B. acknowledge seed-fund support from JARA-FIT.

VI Author contributions

M.H. and B.Z. conceptualized and carried out the DFT calculations. G.M., D.S., M.H., and N.K. developed a spin-dynamics code and M.H. and G.M. performed the spin dynamic simulations. Ch.M. provided the mathematical analysis of the micromagnetic functional. S.B. initiated this work. All authors took part in the analysis and the discussion of the results and contributed to the writing of the paper.

VII Competing financial interests

The authors declare no competing financial interests.